Uniform Eberlein Compactifications of Metrizable Spaces

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provided by Elsevier - Publisher Connector Topology and its Applications 159 (2012) 1691–1694

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Topology and its Applications

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Uniform Eberlein compactiﬁcations of metrizable spaces ∗ Taras Banakh a,b, , Arkady Leiderman c

a Uniwersytet Jana Kochanowskiego w Kielcah, Poland b Ivan Franko National University of Lviv, Ukraine c Ben-Gurion University of the Negev, Israel

article info abstract

MSC: We prove that each metrizable space X (of size |X| c) has a (first countable) uniform 54D35 Eberlein compactification and each scattered metrizable space has a scattered hereditarily 54G12 paracompact compactification. Each compact scattered hereditarily paracompact space 54D30 is uniform Eberlein and belongs to the smallest class A of compact spaces, which 54D20 contains the empty set, the singleton, and is closed under producing the Alexandroff compactification of the topological sum of a family of compacta from the class A. Keywords: © Scattered space 2011 Elsevier B.V. All rights reserved. Metrizable space Scattered compactification Hereditarily paracompact space Uniform Eberlein compact space

Looking for compactifications of metrizable spaces it is natural to search them among compacta having as much prop- erties of metrizable spaces as possible. In this respect, the class of Eberlein compacta fits quite well: Eberlein compacta are Fréchet–Urysohn, they contain metrizable dense Gδ-subsets, their weight coincides with their cellularity, etc. In [2] Arkhangelski˘ı showed that each metrizable space has an Eberlein compactification. Note also that the Eberlein compactification cX of X can be constructed to have dimension dim(cX) = dim(X) [4,6]. The class of Eberlein compacta contains a subclass consisting of uniform Eberlein compacta. Let us recall that a compact space K is (uniform) Eberlein if K is homeomorphic to a compact subspace of a (Hilbert) Banach space endowed with the weak topology, see [3]. The class of uniform Eberlein compact is strictly smaller than the class of Eberlein compacta [3,7]. In this paper we improve the mentioned compactification result of Arkhangelski˘ı [2] constructing a uniform Eberlein compactification for each metrizable space.

Theorem 1. Each metrizable space X has a uniform Eberlein compactiﬁcation.

Proof. Instead of modifying the original construction of Arkhangelski˘ı [2] we shall present an alternative analytic proof. Given a metric space X, apply the classical Dowker result [5, 4.4.K] to ﬁnd an embedding f : X → S to the unit sphere S of a Hilbert space H of suitable density. It is well known that the norm and weak topologies coincide on S.So,S can be considered as a subspace of the closed unit ball B of H endowed with the weak topology. In such a way we obtain an embedding of the metric space X into the metrizable subspace S of the weak unit ball B of H, which is a uniform Eberlein compactum. The closure of X in B will be the desired uniform Eberlein compactiﬁcation of X. 2

Modifying the original approach of Arkhangelski˘ı, we can construct ﬁrst countable uniform Eberlein compactiﬁcations.

* Corresponding author at: Ivan Franko National University of Lviv, Ukraine. E-mail addresses: [email protected] (T. Banakh), [email protected] (A. Leiderman).

Theorem 2. A metrizable space X has a ﬁrst countable uniform Eberlein compactiﬁcation if and only if |X| c.

Proof. The “only if ” part follows from the famous Arkhangelski˘ı’s result [5, 3.1.30] saying that each first countable compact Hausdorff space K has cardinality |K | c. To prove the “if ” part, we shall use Kowalsky’s theorem [5, 4.4.9] asserting that ω each metrizable space X of weight κ embeds into the countable power (Hκ ) of the metric hedgehog Hκ = teα: t ∈[0, 1], α ∈ κ ⊂ l2(κ) with κ spines. Here (eα)α∈κ is the standard orthonormal base of the Hilbert space l2(κ).IfX has weight c,thenX ω embeds into the countable power Hc of Hc. Since the countable product of first countable uniform Eberlein compacta is first countable and uniform Eberlein, it suffices to show that the hedgehog Hc has a first countable uniform Eberlein compactification. This compactification can be constructed explicitly. First fix an injective enumeration {ϕα: α ∈ c} of a topological copy of the Cantor set in [π/6, π/3]. The embedding f : Hc → S of the hedgehog Hc into the unit sphere S of the Hilbert space l2(c + 2) can be defined as follows: f : teα → cos(t)ec+1 + sin(t) cos(ϕα)ec + sin(ϕα)eα for t ∈[0, 1] and α ∈ c.

One can show that the closure of f (Hc) in the weak topology of l2(c + 2) coincides with the set f (Hc) ∪ cos(t)ec+1 + sin(t) cos(ϕα)ec: t ∈[0, 1], α ∈ c homeomorphic to the cone over the Alexandroff duplicate of the Cantor set. The latter set is ﬁrst countable and being a compact subset of (l2(c+2), weak),isuniformEberlein. 2

Remark 1. Independently and simultaneously the result that each metrizable space has a uniform Eberlein compactiﬁcation has been obtained by B.A. Pasynkov [10].

Next, we shall be interested in scattered uniform Eberlein compactifications of scattered metrizable spaces. We recall that a topological space X is scattered if each subspace of X has an isolated point. According to [13] each metrizable scattered space has a compactification homeomorphic to an initial segment of ordinals endowed with the order topology. Unfortunately, such uncountable segments are not Fréchet–Urysohn and hence not Eberlein compact. Nonetheless, scattered metrizable spaces have scattered compactifications which are hereditarily paracompact and uniform Eberlein. The class of scattered hereditarily paracompact compacta is very narrow and admits a constructive description. Namely, it coincides with the smallest class A of compacta, which contains the empty set, the singleton, and the Alexandroff compactifications of topological sums of compacta from the class A. We recall that a compactification X of a regular locally compact space X is Alexandroff if its remainder has size |X \ X| 1.

Theorem 3.

1. Each scattered metrizable space X has a scattered hereditarily paracompact compactiﬁcation K . 2. Each scattered hereditarily paracompact compact space is uniform Eberlein. 3. A compact space X is scattered and hereditarily paracompact if and only if it belongs to the class A.

Remark 2. The class A is not closed with respect to ﬁnite products: the product αℵ1 × αℵ0 of the Alexandroff compacti- ﬁcations of the discrete spaces of size ℵ1 and ℵ0 does not belong to the class A because it is not hereditarily normal, see [5, 2.7.16(a)]. On the other hand, αℵ1 × αℵ0 is a scattered uniform Eberlein compact space. It is known that each scattered Eberlein compact space K is strong Eberlein in the sense that it embeds into the σ -product κ (xα)α∈κ ∈{0, 1} : {α ∈ κ: xα = 0} < ℵ0 of the two-point space for some cardinal κ, see [1].

Remark 3. In light of Theorem 3 it should be mentioned that there are scattered countable regular (and hence hereditarily paracompact) spaces admitting no scattered compactiﬁcation, see [8,9,11,12].

The remaining part of the paper is devoted to the proof of Theorem 3, which is done by transfinite induction using two ordinal functions: the scattered height of a scattered space and the complexity number of a space K ∈ A. (α) = (β) (0) = The scattered height of a scattered space X is defined using the α-th derived sets X β<α(X ) where X X and A denotes the set of non-isolated points of a space A. It is easy to see that a space X is scattered if and only if X(α) is empty for some ordinal α. The smallest ordinal α such that |X(α)| 1iscalledthescattered height of X and is denoted by ht(X). For example, a space X with a unique non-isolated point has ht(X) = 1. T. Banakh, A. Leiderman / Topology and its Applications 159 (2012) 1691–1694 1693 A A = A A Next,weobservethattheclass can be written as the union α α where 0 is the class of all spaces X with |X| 1, and Aα consists of the Alexandroff compactifications of topological sums of compacta from the class A<α = A ∈ A ∈ A β<α β .Thecomplexity number cx(K ) of a space K is the smallest ordinal α such that K α.

Lemma 1. Any compact space K ∈ A is a scattered, hereditarily paracompact, uniform Eberlein and has cx(K ) = ht(K ).

Proof. This lemma will be proved by induction on cx(K ). The assertion is trivial if cx(K ) = 0 (in which case |K | 1). Assume that the lemma is proved for compacta X ∈ A with cx(X)<α. Fix any compact space K ∈ A with cx(K ) = α > 0. { ∈ I}⊂A Then there is a family Ki : i <α such that K is the Alexandroff compactification of the topological sum i∈I Ki . By the inductive assumption, each space Ki is scattered, hereditarily paracompact, uniform Eberlein and has cx(Ki ) = ht(Ki ). Then cx(K ) = sup cx(Ki) + 1 = sup ht(Ki) + 1 = ht(K ). i∈I i∈I To show that K is scattered and hereditarily paracompact, take any infinite subspace X ⊂ K . Being infinite, the set X meets some set Ki .SinceKi is scattered, the intersection X ∩ Ki contains an isolated point which is isolated in X too ∩ (because X Ki is open in X). This proves that K is scattered. ⊂ Next, we show that X is paracompact. This is trivial if X i∈I Ki (because the hereditary paracompactness is preserved by topological sums). So we assume that X contains the compactifying point ∞ of K . Take any open cover U of X and find asetU∞ ∈ U containing the compactifying point ∞.BytheregularityofX, find a closed neighborhood W ⊂ U∞ of ∞.The V \{∞} U hereditary paracompactness of i∈I Ki allows us to find a locally finite open cover of X refining the cover .Then W ={U∞}∪{V \ W : V ∈ V} is a locally finite open cover of X refining the cover U and witnessing that X is paracompact. Finally, we show that K is a uniform Eberlein compactum. For every i ∈ I find an embedding ei : Ki → Hi into an infinite- dimensional Hilbert space endowed with the weak topology. Take a family {Γi}i∈I of pairwise disjoint sets of cardinality | | Γi equal to the density of the Hilbert space Hi . By the classical Riesz theorem, Hi can be identified with the Hilbert space l (Γ ) ={f : Γ → R: f (γ )2 < +∞}. After a suitable affine transformation, we can assume that the image e (K ) does 2 i i γ ∈Γi i i = = not contain the origin 0 of l2(Γi) Hi and lies in the closed unit ball centered at the origin. Let Γ i∈I Γi and observe that the map e : K → l2(Γ ) defined by

∈ ∈ I = ei(x) if x Ki for some i e(x) ∈ \ 0 if x K i∈I Ki is an embedding of K into the Hilbert space l2(Γ ) endowed with the weak topology. This witnesses that K is a uniform Eberlein compact. 2

A topological space X is called strongly zero-dimensional if each open cover U of X has a discrete open reﬁnement V (the latter means that each point of X has a neighborhood meeting at most one set from V). We shall need the following result due to R. Telgarsky [13].

Lemma 2. Each scattered paracompact space is strongly zero-dimensional.

In the following lemma we prove the third statement of Theorem 3. This lemma combined with Lemma 1 implies also the second statement of Theorem 3.

Lemma 3. A compact space X belongs to the class A if and only if X is scattered and hereditarily paracompact.

Proof. The “only if ” part has been proved in Lemma 1. To prove the “if ” part, assume that X is a compact, scattered, hereditarily paracompact space. We need to prove that X ∈ A. This will be done by induction on the scattered height of X. The inclusion X ∈ A is trivial if ht(X) = 0 (in which case X is empty or a singleton). Assume that ht(X) = α > 0 and all scattered hereditarily paracompact spaces with scattered height < α belong to the class A. − It follows that the α-derived set X(α) contains at most one point. If it is empty, then α is not limit and X(α 1) is finite. Being scattered, the compact space X is zero-dimensional. Consequently, we can find a finite cover {Xi: i ∈ I} of X (α−1) consisting of pairwise disjoint clopen subsets such that each set Xi has at most one-point intersection with the set X . | (α−1)| − ∈ A ∈ A It follows that Xi 1 and thus ht(Xi) β 1 < α . By the inductive assumption, each Xi and then X ,being = the Alexandroff compactification of the topological sum i∈I Xi X. Next, we consider the case of a singleton X(α) ={a}. In this case X is the Alexandroff compactification of the locally compact space X \{a}. The space X \{a}, being scattered and paracompact, is strongly zero-dimensional by Lemma 2. This allows us to find a discrete cover {Xi: i ∈ I} of X \{a} that consists of clopen subsets of X \{a} whose closures in X do not contain the compactifying point {a}. Moreover, if the ordinal α is not limit, we may additionally require that each set Xi , − i ∈ I has at most one-point intersection with the set X(α 1) (which has a unique non-isolated point a and hence is closed 1694 T. Banakh, A. Leiderman / Topology and its Applications 159 (2012) 1691–1694

\{ } and discrete in X a ). It follows that each space Xi is compact and has ht(Xi)<α. By the inductive assumption, each A { } \{ } ={ }∪ space Xi belongs to . The discreteness of the family X i in X a implies that X a i∈I Xi can be identified with ∈ A A 2 the Alexandroff compactification of the topological sum i∈I Xi .Consequently,X by the definition of .

It is well known that each ordinal α can be uniquely written as α = β + n(α) for some limit ordinal β and some ﬁnite ordinal n(α). The following lemma combined with Lemma 2 yields (a controlled version of) the ﬁrst statement of Theorem 3.

Lemma 4. Each scattered metrizable space X with scattered height ht(X) = α has a compactiﬁcation K ∈ A with scattered height ht(K ) α + n(α) + 1.

Proof. The proof will be done by induction on α.Ifht(X) = 0, then |X| 1 and K = X is the compactification of X in the class A with ht(K ) = 0 1 = α + n(α) + 1. Now assume that the lemma is proved for scattered metrizable spaces with scattered height < α.LetX be a scattered metrizable space with ht(X) = α.Then|X(α)| 1but|X(β)| > 1forallβ<α. We shall consider separately three cases. (α) =∅ (β) = (α) =∅ = \ (β) 1. α is a limit ordinal and X .Then β<α X X and hence X β<α X X . Being scattered and paracompact, the space X is strongly zero-dimensional by Lemma 2. Consequently, it admits a discrete open cover { ∈ I} \ (β) (β) =∅ ∈ I U i : i such that each set U i lies in X X for some β<α.ThenU i and hence ht(U i ) β<α for all i . ∈ A + + By the inductive assumption, U i has a compactification Ki with ht(Ki ) β n(β) 1 < α.LetK be the Alexandroff = + compactification of the topological sum i∈I Ki . It follows that ht(K ) α α n(α). The discreteness of the cover {U i }i∈I implies that X is homeomorphic to the topological sum

U i ⊂ Ki ⊂ K i∈I i∈I and hence K is the compactification of X. − 2. α is not limit and X(α) =∅.ThenX(α 1) is a closed discrete subspace of X.SinceX is strongly zero-dimensional, (α−1) there is a discrete cover {Xi}i∈ω of X by clopen subsets having at most one-point intersection with the set X .It ∈ I | (α−1)| − follows that for each i we get Xi 1 and thus ht(Xi) α 1 < α. By the inductive assumption, the space Xi ∈ A − + − + = + − has a compactification K i with ht(Ki ) α 1 n(α 1) 1 α n(α) 1. The Alexandroff compactification K A + = of the topological sum i∈I Ki belongs to and has scattered height ht(K ) α n(α).SinceX i∈I Xi , K is a compactification of X. (α) 3. X ={a} is a singleton. Being first countable, X has a decreasing neighborhood base (Wn)n∈ω at x with W 0 = X. Moreover, taking into account that X is zero-dimensional (being scattered and paracompact), we may assume that each (α) set Wn is clopen in X.Foreveryn ∈ ω consider the clopen subspace Yn = Wn \ Wn+1 of X.SinceYn ∩ X =∅,we (α) =∅ = = conclude that Yn and hence β ht(Yn) α.Ifβ α, then by the preceding cases we can find a compactification ∈ A + ∈ A Kn of Yn with ht(Kn) α n(α).Ifβ<α, then by the inductive assumption we can find a compactification Kn + + + of Yn with ht(Kn) β n(β) 1 α n(α). The Alexandroff compactification K of the topological sum n∈ω Kn A ={ }∪ + belongs to , is a compactification of the space X a n∈ω Yn, and has scattered height ht(K ) supn∈ω(ht(Kn) 1) α + n(α) + 1. 2

References

[1] K. Alster, Some remarks on Eberlein compacts, Fund. Math. 104 (1979) 43–46. [2] A. Arkhangelski˘ı, Topological Function Spaces, Kluwer, 1992. [3] Y. Benyamini, T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (2) (1976) 137–141. [4] M. Charalambous, The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces, Fund. Math. 182 (1) (2004) 41–52. [5] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [6] T. Kimura, K. Morishita, On Eberlein compactifications of metrizable spaces, Fund. Math. 171 (3) (2002) 223–234. [7] A.G. Leiderman, G.A. Sokolov, Adequate families of sets and Corson compacts, Comment. Math. Univ. Carolinae 25 (1984) 233–246. [8] V.I. Malykhin, Scattered spaces that have no scattered compact extensions, Mat. Zametki 23 (1) (1978) 127–136. [9] P. Nyikos, Notices Amer. Math. Soc. 21 (6) (1974) A-570, Abstract 716-G5. [10] B.A. Pasynkov, What is a non-metrizable analog of metrizable compacta? (Part I), Topology Appl. 159 (7) (2012) 1750–1760. [11] M. Rajagopalan, Scattered spaces, III, J. Indian Math. Soc. (N.S.) 41 (1977) 405–427. [12] R. Solomon, A space of the form N ∪ (p) with no scattered compactification, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24 (1976) 755–756. [13] R. Telgarsky, Total paracompactness and paracompact dispersed spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 16 (1968) 567–572.